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mathematics
calculus 10th edition

Questions and Answers of Calculus 10th Edition

In Exercises determine the work done by the constant force.A 1200-pound steel beam is lifted 40 feet.
In Exercises determine the work done by the constant force.An electric hoist lifts a 2500-pound car 6 feet.
In Exercises determine the work done by the constant force.A force of 112 newtons is required to slide a cement block 8 meters in a construction project.
In Exercises determine the work done by the constant force.The locomotive of a freight train pulls its cars with a constant force of 9 tons a distance of one-half mile.
In Exercises use Hooke’s Law to determine the variable force in the spring problem.A force of 5 pounds compresses a 15-inch spring a total of 3 inches. How much work is done in compressing the
In Exercises use Hooke’s Law to determine the variable force in the spring problem.A force of 250 newtons stretches a spring 30 centimeters. How much work is done in stretching the spring from 20
In Exercises use Hooke’s Law to determine the variable force in the spring problem.A force of 20 pounds stretches a spring 9 inches in an exercise machine. Find the work done in stretching the
In Exercises use Hooke’s Law to determine the variable force in the spring problem.An overhead garage door has two springs, one on each side of the door. A force of 15 pounds is required to stretch
In Exercises use Hooke’s Law to determine the variable force in the spring problem.Eighteen foot-pounds of work is required to stretch a spring 4 inches from its natural length. Find the work
In Exercises use Hooke’s Law to determine the variable force in the spring problem.Seven and one-half foot-pounds of work is required to compress a spring 2 inches from its natural length. Find the
Neglecting air resistance and the weight of the propellant, determine the work done in propelling a five-ton satellite to a height of (a) 100 miles above Earth(b) 300 miles above Earth
A rectangular tank with a base 4 feet by 5 feet and a height of 4 feet is full of water (see figure). The water weighs 62.4 pounds per cubic foot. How much work is done in pumping water out over the
A cylindrical water tank 4 meters high with a radius of 2 meters is buried so that the top of the tank is 1 meter below ground level (see figure). How much work is done in pumping a full tank of
Use the information in Exercise 11 to write the work W of the propulsion system as a function of the height h of the satellite above Earth. Find the limit (if it exists) of Was h approaches
Neglecting air resistance and the weight of the propellant, determine the work done in propelling a 10-ton satellite to a height of (a) 11,000 miles above Earth(b) 22,000 miles above Earth
A lunar module weighs 12 tons on the surface of Earth. How much work is done in propelling the module from the surface of the moon to a height of 50 miles? Consider the radius of the moon to be 1100
An open tank has the shape of a right circular cone (see figure). The tank is 8 feet across the top and 6 feet high. How much work is done in emptying the tank by pumping the water over the top edge?
Suppose the tank in Exercise 17 is located on a tower so that the bottom of the tank is 10 meters above the level of a stream (see figure). How much work is done in filling the tank half full of
The fuel tank on a truck has trapezoidal cross sections with the dimensions (in feet) shown in the figure. Assume that the engine is approximately 3 feet above the top of the fuel tank and that
Water is pumped in through the bottom of the tank in Exercise 19. How much work is done to fill the tank(a) To a depth of 2 feet?(b) From a depth of 4 feet to a depth of 6 feet?Data from in Exercise
A hemispherical tank of radius 6 feet is positioned so that its base is circular. How much work is required to fill the tank with water through a hole in the base when the water source is at the base?
In Exercises consider a 15-foot hanging chain that weighs 3 pounds per foot. Find the work done in lifting the chain vertically to the indicated position.Repeat Exercise 29 raising the bottom of the
In Exercises find the work done in pumping gasoline that weighs 42 pounds per cubic foot.A cylindrical gasoline tank 3 feet in diameter and 4 feet long is carried on the back of a truck and is used
In Exercises determine which value best approximates the length of the arc represented by the integral. (Make your selection on the basis of a sketch of the arc, not by performing any
In Exercises find the work done in pumping gasoline that weighs 42 pounds per cubic foot.The top of a cylindrical gasoline storage tank at a service station is 4 feet below ground level. The axis of
In Exercises consider a 15-foot hanging chain that weighs 3 pounds per foot. Find the work done in lifting the chain vertically to the indicated position.Take the bottom of the chain and raise it to
In Exercises consider a 20-foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level. Find the work done by the winch in winding up the specified amount of chain.Wind
In Exercises consider a 20-foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level. Find the work done by the winch in winding up the specified amount of chain.Wind
The graphs show the force Fi (in pounds) required to move an object 9 feet along the x-axis. Order the force functions from the one that yields the least work to the one that yields the most work
In Exercises consider a 20-foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level. Find the work done by the winch in winding up the specified amount of chain.Run
In Exercises consider a 20-foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level. Find the work done by the winch in winding up the specified amount of chain.Wind
Verify your answer to Exercise 34 by calculating the work for each force function.Data from in Exercise 34The graphs show the force Fi (in pounds) required to move an object 9 feet along the x-axis.
In Exercises use the integration capabilities of a graphing utility to approximate the work done by a press in a manufacturing process. A model for the variable force F (in pounds) and the distance x
State the definition of work done by a constant force.
State the definition of work done by a variable force.
In Exercises use the integration capabilities of a graphing utility to approximate the work done by a press in a manufacturing process. A model for the variable force F (in pounds) and the distance x
Which of the following requires more work? Explain your reasoning.(a) A 60-pound box of books is lifted 3 feet.(b) A 60-pound box of books is held 3 feet in the air for 2 minutes.
In Exercises use the integration capabilities of a graphing utility to approximate the work done by a press in a manufacturing process. A model for the variable force F (in pounds) and the distance x
Two electrons repel each other with a force that varies inversely as the square of the distance between them. One electron is fixed at the point (2, 4). Find the work done in moving the second
In Exercises use the integration capabilities of a graphing utility to approximate the work done by a press in a manufacturing process. A model for the variable force F (in pounds) and the distance x
In Exercises find the work done by the gas for the given volume and pressure. Assume that the pressure is inversely proportional to the volume.A quantity of gas with an initial volume of 2 cubic feet
In Exercises find the work done by the gas for the given volume and pressure. Assume that the pressure is inversely proportional to the volume.A quantity of gas with an initial volume of 1 cubic foot
The hydraulic cylinder on a woodsplitter has a 4-inch bore (diameter) and a stroke of 2 feet. The hydraulic pump creates a maximum pressure of 2000 pounds per square inch. Therefore, the maximum
A right circular cone is generated by revolving the region bounded by y = hx/r, y = h, and x = 0 about the y-axis. Verify that the lateral surface area of the cone is S = πr √r² + h².
Find the area of the zone of a sphere formed by revolving the graph of y = √9 - x², 0 ≤ x ≤ 2, about the y-axis.
The solid shown in the figure has cross sections bounded by the graph of |x|a + |y|ª = 1, where 1 ≤ a ≤ 2.(a) Describe the cross section when a = 1 and a = 2.(b) Describe a procedure for
Find the volume of the solid of intersection (the solid common to both) of the two right circular cylinders of radius whose axes meet at right angles (see figure). X Two intersecting cylinders Solid
Find the volumes of the solids whose bases are bounded by the circle x² + y2 = 4, with the indicated cross sections taken perpendicular to the x-axis.(a) Squares(b) Equilateral triangles(c)
Find the volumes of the solids whose bases are bounded by the graphs of y = x + 1 and y = x² - 1, with the indicated cross sections taken perpendicular to the x-axis.(a) Squares(b) Rectangles of
Prove that if two solids have equal altitudes and all plane sections parallel to their bases and at equal distances from their bases have equal areas, then the solids have the same volume (see
A cable for a suspension bridge has the shape of a parabola with equation y = kx². Let h represent the height of the cable from its lowest point to its highest point and let 2w represent the total
A draftsman is asked to determine the amount of material required to produce a machine part (see figure). The diameters d of the part at equally spaced points x are listed in the table. The
The Humber Bridge, located in the United Kingdom and opened in 1981, has a main span of about 1400 meters. Each of its towers has a height of about 155 meters. Use these dimensions, the integral in
Find the length of the curve y² = x³ from the origin to the point where the tangent makes an angle of 45° with the x-axis.
A tank on a water tower is a sphere of radius 50 feet. Determine the depths of the water when the tank is filled to one-fourth and three-fourths of its total capacity.
Let C be the curve given by ƒ(x) = cosh x for 0 ≤ x ≤ t, where t > 0. Show that the arclength of C is equal to the area bounded by C and the x-axis.Identify another curve on the interval 0
In Exercises set up the definite integral for finding the indicated arc length or surface area. Then use the integration capabilities of a graphing utility to approximate the arc length or surface
A glass container can be modeled by revolving the graph ofabout the x-axis, where x and y are measured in centimeters. Use a graphing utility to graph the function and find the volume of the
Find the volumes of the solids (see figures) generated if the upper half of the ellipse 9x² + 25y² = 225 is revolved about (a) The x-axis to form a prolate spheroid (shaped like a football)(b)
In Exercises set up the definite integral for finding the indicated arc length or surface area. Then use the integration capabilities of a graphing utility to approximate the arc length or surface
In Exercises set up the definite integral for finding the indicated arc length or surface area. Then use the integration capabilities of a graphing utility to approximate the arc length or surface
In Exercises set up the definite integral for finding the indicated arc length or surface area. Then use the integration capabilities of a graphing utility to approximate the arc length or surface
Consider the equation(a) Use a graphing utility to graph the equation.(b) Set up the definite integral for finding the first-quadrant arc length of the graph in part (a).(c) Compare the interval of
A tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of y= 1/8x² √2 - x and the x-axis (0 ≤ x ≤ 2) about the x-axis, where x and y are measured in
A sphere of radius r is cut by a plane h units above the equator, where h < r. Find the volume of the solid (spherical segment) above the plane.
Consider the graph of y² = x(4 - x)² (see figure). Find the volumes of the solids that are generated when the loop of this graph is revolved about (a) The x-axis(b) The y-axis(c) The line x =
A cone of height H with a base of radius r is cut by a plane parallel to and h units above the base, where h < H. Find the volume of the solid (frustum of a cone) below the plane.
Use the disk method to verify that the volume of a sphere is 4/3πr³, where r is the radius.
The region bounded by y = r² - x², y = 0, and x = 0 is revolved about the y-axis to form a paraboloid. A hole, centered along the axis of revolution, is drilled through this solid. The hole has a
Use the disk method to verify that the volume of a right circular cone is 1/3πr²h, where r is the radiusof the base and h is the height.
Let V₁ and V₂ be the volumes of the solids that result when the plane region bounded by y = 1/x, y = 0, x = 1/4, and x = c (where c > 1/4) is revolved about the x-axis and the y-axis,
A pond is approximately circular, with a diameter of 400 feet. Starting at the center, the depth of the water is measured every 25 feet and recorded in the table (see figure).(a) Use Simpson’s Rule
A storage shed has a circular base of diameter 80 feet. Starting at the center, the interior height is measured every 10 feet and recorded in the table (see figure).(a) Use Simpson’s Rule to
For the metal sphere in Exercise 57, let R = 6. What value of r will produce a ring whose volume is exactly half the volume of the sphere?Data from in Exercise 57A manufacturer drills a hole
Find the area of the zone of a sphere formed by revolving the graph of y = √r² - x²,0 ≤ x ≤ a, about the y-axis. Assume that a < r.
Let a sphere of radius r be cut by a plane, thereby forming a segment of height h. Show that the volume of this segment is 1 Th²(3r - h).
Use the graph to match the integral for the volume with the axis of rotation.(a)(b)(c)(d) b y=f(x) a x = f(y) X
The region in the figure is revolved about the indicated axes and line. Order the volumes of the resulting solids from least to greatest. Explain your reasoning.(a) x-axis(b) y-axis(c) x = 3
Consider the plane region bounded by the graph ofwhere a > 0 and b > 0. Show that the volume of the ellipsoid formed when this region is revolved about the y-axis isWhat is the volume when the
A manufacturer drills a hole through the center of a metal sphere of radius R The hole has a radius r. Find the volume of the resulting ring.
In Exercises consider the solid formed by revolving the region bounded by y = √x, y = 0, and x = 4 about the x-axis.Find the value of in the interval [0, 4] that divides the solid into three parts
In Exercises consider the solid formed by revolving the region bounded by y = √x, y = 0, and x = 4 about the x-axis.Find the value of in the interval [0, 4] that divides the solid into two parts of
A right circular cone is generated by revolving the region bounded by y = 3x/4, y = 3, and x = 0 about the y-axis. Find the lateral surface area of the cone.
The graphs of the functions ƒ₁ and ƒ₂ on the interval [a, b] are shown in the figure. The graph of each function is revolved about the x-axis. Which surface of revolution has the greater
Repeat Exercise 49 for a torus formed by revolving the region bounded by the circle x² + y² = r² about the line x = R, where r < R.Data from in Exercises 49A torus is formed by revolving the
Discuss the validity of the following statements.(a) For a solid formed by rotating the region under a graph about the x-axis, the cross sections perpendicular to the x-axis are circular disks.(b)
In Exercises the integral represents the volume of a solid. Describe the solid. TT J2 y4 dy
In Exercises the integral represents the volume of a solid. Describe the solid. *TT/2 TT S 0 sin² x dx
A region bounded by the parabola y = 4x - x² and the x-axis is revolved about the x-axis. A second region bounded by the parabola y = 4 - x² and the x-axis is revolved about the x-axis. Without
What precalculus formula and representative element are used to develop the integration formula for the area of a surface of revolution?
A torus is formed by revolving the region bounded by the circle x² + y² = 1 about the line x = 2 (see figure). Find the volume of this "doughnut-shaped" solid. -1 -1 y 2 X
What precalculus formula and representative element are used to develop the integration formula for arc length?
In Exercises find the volume generated by rotating the given region about the specified line.R₂ about x = 1 1 0.5 R₁ R2 0.5 y=x² R3 + 1 y = x X
In Exercises use the integration capabilities of a graphing utility to approximate the surface area of the solid of revolution. Function y = sin x Interval [0, π] Axis of Revolution x-axis
In Exercises use the integration capabilities of a graphing utility to approximate the surface area of the solid of revolution. Function y = ln x Interval [1, e] Axis of Revolution y-axis
In Exercises set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the x-axis. y = √√√4x², -1 ≤ x ≤ 1
In Exercises find the volume generated by rotating the given region about the specified line.R₂ about x = 0 1 0.5 R₁ R2 0.5 y=x² R3 + 1 y = x X
Define a rectifiable curve.
In Exercises the integral represents the volume of a solid of revolution. Identify (a) The plane region that is revolved(b) The axis of revolution 2πTS" ( п (4- x)e* dx
In Exercises find the volume generated by rotating the given region about the specified line.R3 about x = 1 1 0.5 R₁ R2 0.5 y=x² R3 + 1 y = x X

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